N=8 superconformal mechanics: direct construction

In the present paper we constructed the supercharges and Hamiltonians for all variants of superconformal mechanics associated with the superalgebras $osp(8\mathrm{\mid }2),\mathfrak{F}(4),osp(4^{\star }\mathrm{\mid }4)$, and $su(1,1\mathrm{\mid }4)$. The fermionic and bosonic fields involved were arranged into generators spanning $so(8),so(7),so(5)\oplus su(2)$ and $su(4)\oplus u(1)$ $R$-symmetry currents of the corresponding superconformal algebras. The bosonic and fermionic parts of these $R$-symmetry generators separately define the constants of motion and form the same algebras. The angular part of the supercharges defining the system have the structure ``$(R-symmetrygenerators)\times fermions$’’ while the angular part of Hamiltonian is just a proper sum of full Casimir operators and its purely bosonic and fermionic parts. We also constructed the explicit embedding of the algebras $so(7),so(5)\times su(2)$, and $su(4)\times u(1)$ into $so(8)$, which provide the possibility to explicitly construct the corresponding supercharges.